Since measuring tubes used in practice generally have the shape of a circular cylinder, a stationary, rotationally symmetric flow profile develops in them. The flow is also laminar, particularly in viscous fluids. Its velocity v is therefore a function of the radial distance r from the axis of the measuring tube: EQU v=v(r) (1)
In a laminar flow, fluid layers of different velocities slide on each other, and due to the friction, a shear stress .tau. is developed between the layers. The characteristic flow behavior of a real fluid can be described by a relationship between the shear stress and the velocity change occurring in the radial direction, i.e., the velocity gradient, also referred to as shear rate V: EQU V=.delta.v/.delta.r, (2)
where .delta. is the known differential operator, and V has the dimension time.sup.-1.
For Newtonian fluids (henceforth denoted by the subscript W), Equation (2) is linear: EQU .tau..sub.w =.epsilon..multidot.V, (3)
where the proportionality constant .epsilon. is the dynamic viscosity and has the dimension (force.multidot.time)/length.sup.2.
For non-Newtonian fluids (henceforth denoted by the subscript nW), Equation (2) is nonlinear: EQU .tau..sub.nW =f(V) (4)
The respective flow behavior of non-Newtonian fluids is represented in a flow diagram, the so-called rheogram, in which the shear stress .tau..sub.nW is plotted against the shear rate V. According to the characteristics shown in FIG. 1, different types of non-Newtonian fluids are distinguished:
Characteristic 1 represents a Bingham plastic; PA1 characteristic 2 represents an intrinsically viscous (shear-thinning) fluid; PA1 characteristic 3 represents a Newtonian fluid; and PA1 characteristic 4 represents a dilatant (shear-thickening) fluid. PA1 a first coil and a second coil disposed diametrically opposite to each other on the outside or in a wall of the measuring tube, PA1 said coils serving to produce a magnetic field cutting across the wall of the measuring tube and the fluid when a coil current flows through the coils; PA1 a first electrode serving to pick off a first potential induced by the magnetic field; PA1 a second electrode serving to pick off a second potential induced by the magnetic field; PA1 a respective radius of the measuring tube at the point of each of the electrodes making a first angle of 60.degree. or a second angle of 45.degree. with the direction of the magnetic field; PA1 a coil-current generator; PA1 a double-pole switch with which the two coils are connectable either in series aiding or in series opposition; and PA1 evaluation electronics which form either a first velocity signal proportional to the average velocity v.sub.m1 from a first potential difference u.sub.k1, picked off the electrodes arranged under the first angle .phi..sub.1 with the coils connected in series aiding and, therefore, generating a magnetic field having the strenght B.sub.k, using the following equation: EQU v.sub.m1 =u.sub.k1 /(3.sup.1/2 B.sub.k R)
In the case of non-Newtonian fluids, the quotient of shear stress and shear rate is not constant. Therefore, the viscosity is not constant, either. As a substitute, the apparent viscosity .epsilon..sub.s and the differential viscosity .epsilon..sub.d are defined, which both depend on the rate of shear.
As shown in FIG. 2, the apparent viscosity .epsilon..sub.s of a given point P of a characteristic in the rheogram is the slope of the secant from the coordinate origin to the point P. The differential viscosity .epsilon..sub.d is the slope of the tangent of the characteristic at the point P.
Hence, EQU .epsilon..sub.s =.tau..sub.nW /V (5) EQU .epsilon..sub.d =.delta..tau..sub.nW /.delta.V (6)
For many technical aplications it is possible to approximate the nonlinear characteristic by means of a simple formula. A common approach is to use Ostwald's and de Waele's power law (cf. a book by J. Ulbrecht, P. Mitschka, "Nicht-newtonsche Flussigkeiten", Leipzig, 1967, page 26): EQU .tau..sub.nW =K.vertline.V.vertline..sup.N-1 .multidot.V, (7)
where the vertical lines signify the absolute value. This absolute-value notation allows for the fact that the shear rate V can also be negative.
In Equation (7), K is the consistency of the fluid, and the exponent N is the flow index. If N=1, the fluid is a Newtonian fluid and K is its (constant) viscosity. If N&lt;1, the fluid is an intrinsically viscous fluid, and if N&gt;1, it is a dilatant fluid.
From the consistency K and the flow index N, the apparent viscosity .epsilon..sub.s and the differential viscosity .epsilon..sub.d can be calculated: EQU .epsilon..sub.s =.tau..sub.nW /V=KV.sup.N-1 ( 8) EQU .epsilon..sub.d =.delta..tau..sub.nW /.delta.V=KNV.sup.N-1 ( 9)
Solving the hydrodynamic differential equations for a fluid for which the power relationship (7) is true and which exhibits a fully developed laminar flow gives the following relation for the flow profile (cf. loc. cit., page 30): EQU v=.function.(r)=v.sub.m (1+3N)/(1+N)[1-(r/R).sup.(1+1/N) ].(10)
where v.sub.m is the average velocity, and R is the inside radius of the measuring tube.